Relating Modules and Representations I am currently studying representation theory and am struggling with the concepts which relate modules and representations. The specific question I am looking at right now is this:

Let $F$ be a field of characteristic zero, let $\sigma=(1,2)$ and $\pi=(1,2,3,4) .$ An $F S_{4}$ -representation $\mathfrak{X}$ is defined by
$$\mathfrak{X}(\sigma)=\left(\begin{array}{rrr}
0 & -1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & -1
\end{array}\right) \quad \text { and } \quad \mathfrak{X}(\pi)=\left(\begin{array}{rrr}
1 & 1 & 1 \\
-1 & 0 & 0 \\
0 & -1 & 0
\end{array}\right)$$
Let $V$ denote the $F S_{4}$ -module corresponding to $\mathfrak{X}$. Prove that the $F S_{4}$ -module $V$ is simple.

But while specific help would obviously be greatly appreciated, I could really do with Idiot's Help to start me off getting my head around the general theory.
 A: Here's help for getting off the ground with general theory that addresses your title question: 
As you probably know, for an $R$ module $M$, all of the module axioms together are equivalent to the existence of a ring homomorphism from $R$ into $End(M)$. When we're talking about representations over fields, $M$ is also (usually) a finite dimensional vector space, so $End(M)=End(F^n)=M_n(F)$ where $n$ is the dimension of the representation as a vector space.
Now group representations are usually first defined as a group homomorphism  $\rho:G\rightarrow GL(n,F)$. This kind of resembles the ring homomorphism above, but right now it's only homomorphism of groups. The trick is that since $\rho:G\rightarrow GL(n,F)\subset M_n(F)$, it extends to be a ring homomorphism from group ring $F[G]\rightarrow M_n(F)$.
At this point you have the extension $\hat{\rho}:F[G]\rightarrow End(M)$, and guess what: that means the extension $\hat{\rho}$ provides an $F[G]$ module structure to $M$!
The connection also works conversely: if you are given an $F[G]$ module, then by just restricting it to $G$ inside of $F[G]$, you get the group homomorphism from $G$ into $GL(n,F)$. This is the correspondence of representations with $F[G]$ modules.

To solve the specific question, you can check if for every $m, n\in M$, there exists $r\in F[G]$ such that $mr=n$ (that would mean $M$ is simple.) But I think Sanchez's suggestion above about characters is better (if you know it). 
